## Stability of a functional equation for square root spirals.(English)Zbl 1016.39020

The authors study a concrete case of the stability (in the sense of Hyers-Ulam-Rassias) of the equation of the square root spiral: $f( \sqrt {r^2 +1}) = f (r) + \arctan \tfrac 1r.\tag{1}$ The main result is the following. If $$f: [1, \infty) \to [0, \infty)$$ satisfies, for all $$r \geq 1$$, $$|f (\sqrt {r^2 +1})- f(r)- \arctan {1 \over r} |$$ $$\leq g(r)$$, where $$g: [1 , \infty) \to [0, \infty)$$ is such that $$G(r) : = \sum_{k=0}^{\infty} g (\sqrt {r^2 + k}) < \infty$$, then there exists a unique solution $$F$$ of (1) such that, for all $$r \geq 1$$, $|F(r)- f(r) |\leq G(r).$

### MSC:

 39B82 Stability, separation, extension, and related topics for functional equations
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### References:

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